We present a new proof of the differentiability of exponential functions. It is based
entirely on methods of differential calculus. No current or recent calculus text gives or
cites a proof of the differentiability that depends only on such elementary tools. Our
proof makes it possible to give a comprehensive treatment of the derivative properties
of exponential and logarithmic functions in that order in differential calculus, building
on the standard introduction to these topics in precalculus courses. This is the logical
order and has considerable pedagogical merit.
Most calculus books defer the treatment of exponential and logarithmic functions to
integral calculus in order to prove differentiability. A few texts introduce these topics in
differential calculus under the heading of “early transcendentals” but defer the proof of
differentiability to integral calculus. Both approaches have serious pedagogical faults,
which are discussed later in this paper.
Our proof that exponential functions are differentiable provides the missing link
that legitimizes the “early transcendentals” presentation.
Preliminaries
We assume that ar has been defined for a > 0 and r rational in a precalculus course
and that the familiar rules of exponents are known to hold for rational exponents. It is
natural to define ax for a > 0 and x irrational as the limit of ar as r → x through the
rationals. In this way, ax is defined for all real x.
Basic properties of ax for real x are inherited by limit passages from corresponding
properties of ar for r rational. These properties include the rules of exponents with
real exponents and
ax is positive and continuous,
ax is increasing if a > 1,
ax is decreasing if a < 1.
It is not especially difficult to justify the definition of ax for x irrational and to
derive the foregoing properties of ax for x real, but there are a lot of small steps. A
program along these lines is carried out by Courant in [2, pp. 69–70]. The general idea
of each step is well within the grasp of students in typical calculus classes. However,
just as properties of ar with r rational are routinely stated without proof, it is better to
give just an overview of the basic properties of ax with x real, illustrated with graphs,
and move on to the question of differentiability, which is more central to differential
calculus.
A more complete development, beginning with the derivation of properties of ar
with r rational, might be given in an honors class. The properties can be extended to
ax with x real with the aid of the density of the rationals in the reals and the squeeze
laws for limits. The conclusion that ax with a > 1 is increasing also relies on the
following proposition which should seem evident from graphical considerations:
If f is a continuous function on a real interval I
and f is increasing on the rational numbers in I,
then f is increasing on I.
The same proposition will provide a key step in the proof that ax is differentiable.
Henceforth, we restrict our attention to properties of ax with a > 1. Corresponding
properties of ax with 0 < a < 1 follow from ax = (1/a)
−x .
The differentiability of ax
Consider an exponential function ax with any a > 1. In order to prove that ax is differentiable
for all x, the main task is to prove that it is differentiable at x = 0. Our proof
of this depends only on methods of differential calculus. It is motivated by the fact
that the graph of ax (see Figure 1) is concave up, even though this fact is not assumed
a priori
 |
| figure 1 |
Graph of ax with B = (x, ax ) and C = −x, a−x for x > 0
In Figure 1, imagine that x → 0 with x > 0 and x decreasing. Then B and C slide
along the curve toward A. The upward bending of the curve seems to imply that
slope AB decreases, slope AC increases,
and slope AB − slope AC → 0.
It follows that the slopes of AB and AC approach a common limit, which is the slope of
the tangent line T in Figure 1 and the derivative of f (x) = ax at x = 0. This geometric
argument will be made rigorous.
The curve in Figure 1 is actually the graph of f (x) = 2x . The following table gives
values of the slopes of AB and AC rounded off to two decimal places. It appears that
the slopes of AB and AC approach a common limit, which is f
(0) = slope T ≈ 0.7.
x 1 1/2 1/4 1/8 1/16 1/32
slope AB 1 .83 .76 .72 . 71 .70
slope AC .50 .59 .64 . 66 .67 .69
With this preparation, we are ready to prove that f (x) = ax is differentiable at
x = 0. The foregoing geometric description of the proof and the numerical evidence
should be informative and persuasive to students, even if they do not follow all the
details of the argument.
Theorem 1. Let f (x) = ax with any a > 1. Then f is differentiable at x = 0 and
f
(0) > 0.
Proof. To express our geometric observations in analytic terms, let
m(x) = f (x) − f (0)/x − 0
= ax − 1/x
.
In Figure 1, x > 0 and
slope AB = m(x),
slope AC = m(−x).
We shall prove that, as x → 0 with x > 0 and x decreasing, m(x) and m(−x) approach
a common limit, which is f
(0).
To begin with, m(x) is continuous because ax is continuous. The crux of the proof,
and the only tricky part, is to show that
m(x) is increasing on (0,∞) and (−∞, 0).
We give the proof only for (0,∞) since the proof for (−∞, 0) is essentially the same.
We show first that m is increasing on the rationals in (0,∞). Fix rational numbers r
and s with 0 < r < s and let a vary with a ≥ 1. Define
g(a) = m(s) − m(r ) =as -1/s = ar-1/r
.
Then g(a) is continuous for a ≥ 1 and
g
(a) = as -1-ar-1>0 for a>0
Thus, g(a) increases as a increases and g(a) > g(1) = 0 for a > 1, so
m(r) < m(s) for 0 < r < s.
Thus, m(x) is continuous on (0,∞) and m(x) increases on the rational numbers in
(0,∞). As noted earlier, this implies that m(x) increases on (0,∞). The argument for
the interval (−∞, 0) is similar.
For x > 0,
m(−x) = m(x)a−x ,
0 < m(−x) < m(x),
0 < m(x) − m(−x) = m(x) 1 − a−x .
Let x → 0 with x decreasing. Then
m(x) decreases, m(−x) increases, m(x) − m(−x) → 0.
It follows that m(x) and m(−x) approach a common limit as x → 0, which is f
(0).
Furthermore, 0 < m(−x) < f '(0) < m(x) for x > 0, which implies that f ' (0) > 0.
We believe that this proof is new. We have been unable to find any other proof
that depends only on methods of differential calculus. Theorem 1 and familiar reasoning give the principal result on the differentiability
special refrence to www.maa.org
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